Using an electric mixer on medium-high speed, beat eggs and brown sugar in a large bowl until more than tripled in volume and mixture holds a ribbon for several seconds when drizzled over itself, 5–7 minutes. Beat in vanilla.

Reduce mixer speed to low. Add flour mixture in 3 additions, alternating with carrot mixture in 3 additions, to egg mixture, beating well after each addition.

Divide batter between prepared pans. Bake cake until lightly browned across the top, a tester inserted into the center comes out clean, and the top springs back when gently poked, 33–36 minutes. Let cool 10 minutes. Carefully run a knife around edges of pans to release cake, then invert onto a wire rack. Let cool completely.

Frosting

Using electric mixer on medium-high speed, beat cream cheese and butter in a large bowl, scraping down as needed, until very smooth, about 2 minutes. Reduce mixer speed to low and add powdered sugar. Beat until combined. Add yogurt, vanilla, and salt and increase speed to medium-high. Beat, scraping down occasionally, until light and airy, about 4 minutes.

Chill 10 minutes if needed to stiffen slightly to a spreadable consistently.

Arrange one cake round domed side down on a large plate or platter. Cover top and sides with one-third of frosting. Top with remaining cake layer, domed side up. Coat top and sides with remaining frosting.

*Recipe may not look exactly like the picture, picture is just used as a reference.*